Prey and predator are the important factor of the ecosystem. Generally, it is considered that prey–predator models depends on time and it is only required nonlinear system of equations for its dynamical study. But, it is observed that such species can move from one to place to another and in such a way there is a need of nonlinear equations which also depends on spatial as well. The stochastic prey–predator system are investigated numerically and analytically. The proposed stochastic NSFD is used for numerical study; it is consistent with given system and its linear stability analysis showed that it is unconditionally stable. There are two equilibria one is predator free and second is coexistence equilibrium. These equilibria are successfully gained in the numerical case. Extended generalized Riccati equation mapping method is applied for analytical study. The obtained solutions are of the form rational, hyperbolic, and trigonometric. For the comparative study, the unique physical problems are developed and their simulations are drawn for various choices of the parameters. The graphical behavior depicts the efficacy of our study.

In this study, we investigate the abundant soliton solutions for the time-fractional stochastic Gray-Scot (TFSGS) model analytically. The Gray-Scot model is considered under the influence of M-truncated derivative and multiplicative time noise. This is a reaction–diffusion chemical concentration model that explains the irreversible chemical reaction process. The M-truncated derivative is applied for the fractional version while Brownian motion is taken in the sense of time noise. The novel mathematical technique is used to obtain the abundant families of soliton solutions. These solutions are explored in the form of shock, complicated solitary-shock, shock-singular, and periodic-singular types of single and combination wave structures. During the derivation, the rational solutions also appear. Moreover, we use MATHEMATICA 11.1 tools to plot our solutions and exhibit several three-dimensional, two-dimensional, and their corresponding contour graphs to show the fractional derivative and Brownian motion impact on the soliton solutions of the TFSGS model. We show that the TFDGS model solutions are stabilized at around zero by the multiplicative Brownian motion. These wave solutions represent the chemical concentrations of the reactants.